Ah, OK - I think there's probably confusion on ">" vs. "<". Two stopovers 24+ hrs in the continent of origin, of which only 1 could be in the country of origin before an intercontinental flight, and of which only 1 could be in the country of origin upon re-entering the continent of origin at the end of the circle.

I don't think LAX-RDU-LAX is possible in a turnaround; indeed I don't see RDU-LAX in the AA timetable at all; perhaps DownUnderFlyer could mention the dates. These "direct" flights come and go and are typically asymmetrical (e.g. LAX-RDU, RDU-SEA.) When the OP goes to ticketing if they're not there in the time frame for the RTW they won't be allowed.